The first part of the proof essentially follows the argument given in Ref. [1]} Sec. VIIIA and reproduced in Ref. [2]} Sec. IVA, slightly adapted to use the fewer assumptions of the statement. We have that
\(1= p(M_k|P_k) = \int _{S_k} d \lambda \mu _k(\lambda ) \xi _{k}(\lambda ), \quad k=s,s^{\prime },\) 
